Preprints

15. Canonical Stratifications along Bisheaves

A theory of bisheaves has been recently introduced to measure the homological stability of fibers of maps to manifolds. A bisheaf over a topological space is a triple consisting of a sheaf, a cosheaf, and compatible maps from the stalks of the sheaf to the stalks of the cosheaf. In this note we describe how, given a bisheaf constructible (i.e., locally constant) with respect to a triangulation of its underlying space, one can explicitly determine the coarsest stratification of that space for which the bisheaf remains constructible.

14. Equivariant Simplicial Reconstruction

We introduce and analyze parallelizable algorithms to compress and accurately
reconstruct finite simplicial complexes that have non-trivial automorphisms. The compressed
data – called a complex of groups – amounts to a functor from (the poset of simplices in)
the orbit space to the 2-category of groups, whose higher structure is prescribed by
isomorphisms arising from conjugation. Using this functor, we show how to algorithmically recover
the original complex up to equivariant simplicial isomorphism. Our algorithms are derived
from generalizations (by Bridson-Haefliger, Carbone-Rips and Corson, among others) of the
classical Bass-Serre theory for reconstructing group actions on trees.

13. Local Cohomology and Stratification

We outline an algorithm to recover the canonical (or, coarsest) stratification of a given regular CW complex into cohomology manifolds, each of which is a union of cells. The construction proceeds by iteratively localizing the poset of cells about a family of subposets; these subposets are in turn determined by a collection of cosheaves which capture variations in cohomology of cellular neighborhoods across the underlying complex. The result is a finite sequence of categories whose colimit recovers the canonical strata via (isomorphism classes of) its objects. The entire process is amenable to efficient distributed computation.

(2019)

12. Discrete Morse Theory and Localization

Incidence relations among the cells of a regular CW complex produce a 2-category of entrance paths whose classifying space is homotopy-equivalent to that complex. We show here that each acyclic partial matching on (the cells of) such a complex corresponds precisely to a homotopy-preserving localization of the associated entrance path category. Restricting attention further to the full localized subcategory spanned by critical cells, we obtain the discrete flow category whose classifying space is also shown to lie in the homotopy class of the original CW complex. This flow category forms a combinatorial and computable counterpart to the one described here by Cohen, Jones and Segal in the context of smooth Morse theory.

(2018)

11. Persistence Paths and Signature Features in Topological Data Analysis

We introduce a new feature map for barcodes that arise in persistent homology computation. The main idea is to first realize each barcode as a path in a convenient vector space, and to then compute its path signature which takes values in the tensor algebra of that vector space. The composition of these two operations — barcode to path, path to tensor series — results in a feature map that has several desirable properties for statistical learning, such as universality and characteristicness, and achieves state-of-the-art results on common classification benchmarks.

10. Discrete Morse Theory and Classifying Spaces

The aim of this paper is to develop a refinement of Forman’s discrete Morse theory. To an acyclic partial matching µ on a finite regular CW complex X, Forman introduced a discrete analogue of gradient flows. Although Forman’s gradient flow has been proved to be useful in practical computations of homology groups, it is not sufficient to recover the homotopy type of X. Forman also proved the existence of a CW complex which is homotopy equivalent to X and whose cells are in one-to-one correspondence with the critical cells of µ, but the construction is ad hoc and does not have a combinatorial description. By relaxing the definition of Forman’s gradient flows, we introduce the notion of flow paths, which contains enough information to reconstruct the homotopy type of X, while retaining a combinatorial description. The critical difference from Forman’s gradient flows is the existence of a partial order on the set of flow paths, from which a 2-category C(µ) is constructed. It is shown that the classifying space of C(µ) is homotopy equivalent to X by using homotopy theory of 2-categories. This result may be also regarded as a discrete analogue of the unpublished work of Cohen, Jones, and Segal on Morse theory from the early 90’s.

(2017)

09. Topological Signals of Singularities in Ricci Flow

We implement methods from computational homology to obtain a topological signal of
singularity formation in a selection of geometries evolved numerically by Ricci flow. Our approach,
based on persistent homology, produces precise, quantitative measures describing the behavior of
an entire collection of data across a discrete sample of times. We analyze the topological signals of
geometric criticality obtained numerically from the application of persistent homology to models
manifesting singularities under Ricci flow. The results we obtain for these numerical models suggest
that the topological signals distinguish global singularity formation (collapse to a round point) from
local singularity formation (neckpinch). Finally, we discuss the interpretation and implication of
these results and future applications.

Status: Appeared in the open-source MDPI journal Axioms, official publisher version here.

08. Higher Interpolation and Extension for Persistence Modules

The use of topological persistence in contemporary data analysis has provided considerable impetus for investigations into the geometric and functional-analytic structure of the space of persistence modules. In this paper, we isolate a coherence criterion which guarantees the extensibility of non-expansive maps into this space across embeddings of the domain to larger ambient metric spaces. Our coherence criterion is category-theoretic, allowing Kan extensions to provide the desired extensions. As a consequence of such “higher interpolation”, it becomes possible to compare Vietoris-Rips and Cech complexes built within the space of persistence modules.

(2016)

07. Discrete Morse Theory for Computing Cellular Sheaf Cohomology

Sheaves and sheaf cohomology are powerful tools in computational topology, greatly generalizing persistent homology. We develop an algorithm for simplifying the computation of cellular sheaf cohomology via (discrete) Morse-theoretic techniques. As a consequence, we derive efficient techniques for computation of (ordinary) cohomology of a cell complex.

(2015)

06. A Topological Measurement of Protein Compressibility

Given X-ray crystallography data of a protein molecule from the
PDB, we build a van der Waal
weighted alpha
shape representation of that protein molecule by constructing cells around each atom center.
Thus, to each protein we assoicate a set of persistence diagrams (one for each dimension).
Using elementary physical principles, we identify certain structural features of molecules that are
conjectured to impact compressibility. A simple parameter search through the persistence diagrams
isolates these features and provides a robust measure which exhibits remarkable linear correlation
with experimentally computed protein compressibility.

(2014)

05. Reconstructing Functions from Random Samples

From a sufficiently large point sample lying on a compact Riemannian submanifold of Euclidean space, one can construct a simplicial complex which is homotopy-equivalent to that manifold with high confidence. We describe a corresponding result for a Lipschitz-continuous function between two such manifolds. That is, we outline the construction of a simplicial map which recovers the induced maps on homotopy and homology groups with high confidence using only finite sampled data from the domain and range, as well as knowledge of the image of every point sampled from the domain. We provide explicit bounds on the size of the point samples required for such reconstruction in terms of intrinsic properties of the domain, the co-domain and the function. This reconstruction is robust to certain types of bounded sampling and evaluation noise.

We provide explicit and efficient algorithms based on discrete Morse theory to compute homology of a very general class of complexes.
A set-valued map of top-dimensional cells between such complexes is a natural discrete approximation of an underlying (and possibly unknown)
continuous function, especially when the evaluation of that underlying function is subject to measurement errors. We introduce a new Morse theoretic
algorithm for deriving chain maps from these set-valued maps, and hence an effective scheme for computing the map induced on homology by
the approximated continuous function.

03. Simplicial Models and Topological Inference in Biological Systems

This article is a user's guide to algebraic topological methods for data analysis
with a particular focus on applications to datasets arising in experimental biology.
We begin with the combinatorics and geometry of simplicial complexes and outline
the standard techniques for imposing filtered simplicial structures on a general
class of datasets. From these structures, one computes topological statistics of
the original data via the algebraic theory of (persistent) homology. These
statistics are shown to be computable and robust invariants of the shape
underlying a dataset. Finally, we showcase some appealing instances of
topology-driven inference in biological settings -- from the detection of
a new type of breast cancer to the analysis of various neural structures.

(2013)

02. Geometry in the Space of Persistence Modules

We study the geometry of the space of persistence modules and diagrams, with special attention to Cech and Rips complexes. The metric structures are determined in terms of interleaving maps (of modules) and matchings (between diagrams). We show that the relationship between the Cech and Rips complexes is governed by the relationship between the corresponding interleavings and matchings.

We introduce an efficient preprocessing algorithm to reduce the number of cells in a filtered cell complex while preserving its persistent homology groups. The technique is based on an
extension of combinatorial Morse theory from complexes to filtrations of cell complexes.

This paper provides the theoretical basis for the Perseus software project designed to compute persistent homology of various types of filtrations.

(2012)

00. Discrete Morse Theory for Filtrations

My Ph.D. dissertation from October 2012. The main results presented here are from the paper titled "Morse theory for filtrations and efficient computation of persistent homology". The final chapter outlines a bonus application: the filtered Morse theory may be used towards simplifying the construction of long exact sequences via the Zigzag lemma. Here is Rutgers' official copy of the document, which has been formatted with awful (but mandatory) double-spacing, to say nothing of the gigantic margins into which many marvelous proofs would fit.